Theorem mobii 2117

Description: Formula-building rule for "at most one" quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)

Hypothesis

Ref	Expression
mobii.1	⊢ (𝜓 ↔ 𝜒)

Assertion

Ref	Expression
mobii	⊢ (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)

Proof of Theorem mobii

Step	Hyp	Ref	Expression
1		mobii.1	. . . 4 ⊢ (𝜓 ↔ 𝜒)
2	1	a1i 9	. . 3 ⊢ (⊤ → (𝜓 ↔ 𝜒))
3	2	mobidv 2116	. 2 ⊢ (⊤ → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
4	3	mptru 1407	1 ⊢ (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)

Syntax hints:   ↔ wb 105  ⊤wtru 1399  ∃*wmo 2081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-eu 2083  df-mo 2084

This theorem is referenced by:  moaneu  2157  moanmo  2158  2moswapdc  2171  2exeu  2173  rmobiia  2735  rmov  2834  euxfr2dc  3002  rmoan  3017  2rmorex  3023  mosn  3725  dffun9  5381  funopab  5387  funco  5392  funcnv2  5416  funcnv  5417  fun2cnv  5420  fncnv  5422  imadif  5436  fnres  5475  ovi3  6191  oprabex3  6322  axaddf  8183  axmulf  8184  frecuzrdgtcl  10774  frecuzrdgfunlem  10781  fsum3  12073